The Boussinesq system describes the movement of incompressible fluid under the influence of gravitational forces, we refer to [1]} for more applications in fluid mechanics. In the three-dimensional case, the inhomogeneous incompressible Boussinesq system is of the form
\(\left\lbrace \begin{array}{lll}\medskip \partial _t \rho + u\cdot \nabla \rho =0, \qquad (t,x)\in \mathbb {R^+}\times \mathbb {R}^3,\\\medskip \rho \partial _t \theta + \rho (u\cdot \nabla \theta )-\Delta \theta =0,\\\medskip \rho \partial _t u+ \rho (u\cdot \nabla u)-\Delta u+ \nabla \Pi =\rho \theta e_3,\\\medskip \mathrm {div} u=0,\end{array}\right.\) 
